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CS 352: Computer Graphics Chapter 5: Viewing Chapter 5 - 2 Interactive Computer Graphics Overview Specifying the viewpoint Specifying the projection Types of projections Viewing APIs Walking through a scene Projections and shadows Chapter 5 - 3 Interactive Computer Graphics How do cameras work? Chapter 5 - 4 Interactive Computer Graphics Synthentic camera model 1. Camera is placed at a location, pointed in a direction (modeling matrix) 2. 3D points are flattened onto the viewing plane (projection matrix) What do we need to know about the camera (real or synthetic)? Chapter 5 - 5 Interactive Computer Graphics Synthetic camera parameters Position of camera Direction it is pointed [look vector] Angle of film to look vector [view plane normal] Rotation around viewing direction [up vector] Height angle (zoom setting) [fovy] Aspect ratio of "film" (width/height) Front and back clipping planes Focal length Field of view Shutter speed Chapter 5 - 6 Interactive Computer Graphics Chapter 5 - 7 Interactive Computer Graphics Perspective distortion How would you film dizziness? Vertigo effect [2] Chapter 5 - 8 Interactive Computer Graphics Projections Basic Elements: Objects, viewer Projection plane Projectors Center of projection Direction of projection (DOP) Basic Types Perspective Parallel (COP at infinity) Chapter 5 - 9 Classical viewing Interactive Computer Graphics Chapter 5 - 10 Interactive Computer Graphics Orthographic projection Orthographic: parallel projection with projectors perpendicular to the projection plane. Often used as front, side, top views for 3D design Importance: preservation of distance and angle Often used for top, front, and size views, e.g. in a modeling program or working drawing Chapter 5 - 11 Interactive Computer Graphics Perspective projection Perspective projections: projectors converge at COP Classical perspective views: 1, 2, and 3-point (1, 2, or 3 vanishing points) Difference: how many of the principle axes of the object are parallel to projection plane (I.e., depends on relationship of object to viewing frame) Chapter 5 - 12 Interactive Computer Graphics 1. Position the camera By default, camera is at origin, looking in –z dir To “move the camera”, set up a modelview matrix that moves objects that are drawn Ignore Z-coordinate when drawing E.g. dimetric view? modelview = identity translate(0,0,-d) rotate(-45,<0,1,0>); Chapter 5 - 13 Interactive Computer Graphics Exercise: look from +x axis How would you change the camera to generate a view down the +x axis to origin? Do this before displaying objects: modelview = identity; translate(0, 0, -d); rotate(-90, [0, 1, 0]); Chapter 5 - 14 Interactive Computer Graphics Exercise: front/top view How would you change the camera to generate a view from (0, 10, 10) to origin? Do this before displaying objects: modelview = identity; translate(0,0,-14.14); rotate(45, [1, 0, 0]); Chapter 5 - 15 Interactive Computer Graphics Helper function: lookAt Most 3D toolkits let you position the camera by setting eyepoint, lookpoint, and up direction lookAt(Xeye, Yeye, Zeye, Xat, Yat, Zat, Xup, Yup, Zup): Effect: set the modelview matrix Chapter 5 - 16 Interactive Computer Graphics Rolling your own lookAt How could you write your own lookAt function? lookAt(Xeye, Yeye, Zeye, Xat, Yat, Zat, Xup, Yup, Zup): Chapter 5 - 17 Interactive Computer Graphics Defining a lookAt function lookAt(Xeye, Yeye, Zeye, Xat, Yat, Zat, Xup, Yup, Zup): translate <Xat, Yat, Zat> to origin rotate so that <Xeye, Yeye, Zeye> points in the Z direction normalize <Xeye, Yeye, Zeye> trackball-like rotation to <0,0,1> rotate so <Xup, Yup, Zup> is <0,1,0> trackball-like rotation Chapter 5 - 18 Interactive Computer Graphics Camera API 2: uvn frame Camera parameters: VRP: view reference point, a point on the image plane VPN: view plane normal (n) VUP: vector in up direction (also need viewing direction, if not VPN) Result: viewing coordinate system, u-v-n. v = projection of VUP onto image plane u=vxn u, v axes: coordinates in the image plane n axis: normal to image plane Chapter 5 - 19 Interactive Computer Graphics Camera API 3: roll, pitch, yaw Specify location + orientation: roll, pitch, yaw Chapter 5 - 20 Interactive Computer Graphics 2. Specify projection Once we have located and pointed the camera along the –z axis, we still need to specify the lens (projection). Chapter 5 - 21 Interactive Computer Graphics Parallel projection We’re already looking along the –z axis Set z=0 for all points (or ignore z coordinate when rendering) é ê ê ê ê ë 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ù ú ú ú ú û Chapter 5 - 22 Interactive Computer Graphics Parallel projection View volume is generally specified with clipping planes: e.g. glOrtho(xmin, xmax, ymin, ymax, near, far) Z clipping planes at –near and –far Chapter 5 - 23 Interactive Computer Graphics Perspective projection Need to build appropriate perspective projection matrix into vertex shader What kind of transformation would this be? Chapter 5 - 24 Interactive Computer Graphics Perspective projections COP at origin Looking in –z direction Projection plane in front of origin at z=d Chapter 5 - 25 Interactive Computer Graphics Foreshortening By similar triangles in previous image, we see that x x and similarly for y. p z d 1 0 0 0 0 1 0 0 0 0 1 Using the perspective matrix we get p’ = x y z z d T Adding divide-by-w to the graphics pipeline gives the correct result. 1 d 0 0 0 0 Chapter 5 - 26 Interactive Computer Graphics Perspective Frustum Perspective viewing region is a “frustum”: Viewplane normally coincides with front clip plane Chapter 5 - 27 Interactive Computer Graphics Camera APIs In raw OpenGL ES, you “position the camera” by programming a vertex shader to apply a modelview matrix Frameworks provide functions to build a viewing matrix for you, using a “camera API” Example: perspectiveCamera(FOV, aspect, near, far) Chapter 5 - 28 Interactive Computer Graphics Perspective projection 3D graphics toolkits provide tools for specifying a perspective projection, e.g. Chapter 5 - 29 Interactive Computer Graphics Shadows How can one generate shadows in a scene using interactive graphics techniques? In general it's hard, not supported in standard graphics pipeline—you need to know where everything is globally to render a point locally Special techniques let you “fake it” Chapter 5 - 30 Interactive Computer Graphics Projections and shadows Projections can be used to generate simple shadow polygons Light (xl, yl, zl) Translate light to origin Project down y axis [M] Translate back 1 0 0 0 0 1 0 1 yl 0 0 0 0 1 0 0 0 Chapter 5 - 31 Interactive Computer Graphics Simple Shadow in OpenGL GLfloat m[16]; //projection matrix for (int i=0; i<16; i++) m[i]=0; m[0]=m[5]=m[10]=1; m[7] = -1/yl; glBegin(); [draw polygon normally]; glEnd(); glMatrixMode(GL_MODELVIEW); glPushMatrix; //save state glTranslatef(xl, yl, zl); glMultMatrix(m); glTranslatef(-xl, -yl, -zl); glColor3fv(shadow_color); [draw polygon] glEnd(); glPopMatrix(); Chapter 5 - 32 Interactive Computer Graphics Stereo Viewing In our stereo setup, we need two images (4x3 size ratio), side-by-side We want to render perspective views from viewpoints (say) 3 inches apart How to set up the views?* * “Simple, Low-Cost Stereographics: VR for Everyone,” J. Zelle & C. Figura, Proc. SIGCSE 2004 p. 348. Chapter 5 - 33 Interactive Computer Graphics Direct both eyes at the same point? Chapter 5 - 34 Interactive Computer Graphics Direct both eyes in parallel? Chapter 5 - 35 Interactive Computer Graphics Parallel views and asymmetric frustum Chapter 5 - 36 Interactive Computer Graphics Stereo viewing: // set up the projection transformation // focalLength is distance to screen (objects // closer will float in front of screen) top = eyeSeparation / 2.0 * (near / focalLength); glFrustum(-right+off, right+off, -top, top, near, far); // now set up the model-view transformation // right is a unit vector in right direction viewpoint = viewpoint – right * eyeOffset; center = center – right * eyeOffset; gluLookAt(viewpoint[X],viewpoint[Y],viewpoint[Z], center[X], center[Y], center[Z], up[x], up[y], up[z]); Chapter 5 - 37 Interactive Computer Graphics Drawing left and right views //create window int width=400; int height=300; glutInitWindowSize(2*width, height); //------------------------------------// draw left image glViewport (0, 0, width, height); // set up projection and modeling matrices // render image // draw right image glViewport (width, 0, width, height); // set up projection and modeling matrices // render image Chapter 5 - 38 Interactive Computer Graphics Walking through a scene How to animate viewer motion through a scene? [Demo] Assume viewer’s height is fixed; looking direction is in y=6 plane Store viewer’s location (x,6,z) and orientation (θ). Update appropriately with user commands LookAt( x, y, z, x + cos(θ), y, z + sin(θ), 0, 1, 0); Chapter 5 - 39 Credits 1. (Pinhole camera): Wikipedia. 5. Synthetic camera parameters: Liz Marai, Pitt Demos Musical solar system Interactive Computer Graphics Chapter 5 - 40 Interactive Computer Graphics Chapter 5 - 41 Interactive Computer Graphics